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dueling dualities

Every year, Edge founder John Brockman asks a big question and fields responses from big-thinking scientists and writers -- including the Spousal Unit (and 155 others, so it's not something you can peruse quickly in full). This year's question is, "What scientific concept would improve everybody's cognitive toolkit?" I probably would have picked something pathetically obvious, like, say, entropy, or confirmation bias -- because if more people were self-aware enough not to fall into knee-jerk denialism by cherry picking the "evidence" to support pre-existing frameworks, the world would be a much better place. So I was impressed when New Scientist's Amanda Gefter went boldly into the arcane landscape of theoretical physics and chose the concept of duality. I'd relegated this fascinating notion to the back burners of my mind for the last few years; it was nice to be reminded of just how seminal dualities are to modern physics, which thrives on uncovering hidden connections.

No doubt a few of you are wondering just what I'm talking about. Dualities in physics refer to theoretical models that appear to be different but can be shown to describe exactly the same thing. It’s a bit like how ice, water and vapor are three different phases of the same chemical substance, except a duality looks at the same phenomenon in two different ways that are inversely related. For instance, subatomic particles like photons and electrons also exhibit wave-like behavior, but uncertainty dictates that we can’t see both of these aspects at the same time. The more accurately we observe an electron's particle nature, the less we can see the wave-like nature of the object, and vice versa. This is a duality. The particle/wave identities are inversely related, different “faces” of the same underlying reality.

String theory has its own set of dualities. As Amanda points out in her Edge essay, once upon a time there were several competing dimensional theories – five, in fact – and for a long time, no one knew which version was correct. Then Edward Witten stepped in and demonstrated that the five different string theories weren’t so contradictory after all. He united all five under a single theoretical umbrella that he dubbed M theory.

M theory indicates that mathematically, the five versions of string theory are merely five different ways of looking at the same thing. Each iteration of string theory is connected in some way to another through various dualities, in an intricate web of interconnections that ultimately links all five to each other and to M theory. Stringy dualities can help physicists simplify difficult calculations through a kind of “bait and switch” approach; that's what physicists mean when they talk about the "non-perturbative properties" of string theory.

Standard perturbation theory is a calculating method that makes approximations to get a rough answer and then refines it bit by bit, according to how given physical systems are known to behave. Physicists employ this kind of approach all the time, say, when determining the trajectory of a satellite. They use Newton’s laws for the initial calculation, and then make small refinements by calculating the effects of other factors that might influence the trajectory: pressure from the solar wind, for instance, or the effects of heating on one side of the satellite.

Ideally, there should be only small discrepancies between the original calculation and the subsequent refinements. But sometimes the systems are too complicated, such when physicists try to calculate the highly turbulent air flow patterns of a tornado, or the properties of a rapidly rotating wormhole. Their perturbative approximations break down because they are dealing with sudden, large changes to the original value, instead of small, predictable increments. They can't use a perturbative approach, because the numbers involved would be too large. Stringy dualities could help simplify these messy calculations.

For instance, string theory has something called T-duality. It describes an inverse mirror relationship between two of the five permutations of string theories. Here’s how it works: if the radius (r) of a circular area of space has a large value (r = 1000) in one of the five versions of string theory, then we are dealing with small amounts of energy and it is relatively simple to calculate. However, that value will be inverted, and therefore small (r = 1/1000) in one of the other versions, and we would be dealing with large energies. This is a large perturbation, and much more difficult to calculate. Because both theories describe the same underlying physics, scientists can use the dual theory where the value is small to calculate the quantity, then plug it back into the original theory to complete the calculation.

Dualities are a type of symmetry. The concept is a natural extension of the many different kinds of symmetry we see every day around us in the physical world. For instance, rotate a snowflake by 60 degrees, and you’ll find it looks exactly the same. This is spatial symmetry. A second type of symmetry occurs when one shuffles a series of similar objects, like a shell game, where the player must guess under which shell a marble might be found after the three are randomly shuffled in quick succession. Regardless of where the marble turns out to be, mathematically, there are six different ways in which three identical objects can be interchanged. Quantum chromodynamics (QCD) describes the strong nuclear force and the way various quarks interact with each other. There are quarks of three different “colors” that can be randomly interchanged, just like the shells, so those quarks share a similar internal symmetry.

Supersymmetry extends this interchangeable shuffling to incorporate all known subatomic particles. Not all potential couplings are feasible in the current standard model. Fermions (the particles that make up matter) and bosons (messenger particles that carry fundamental forces) can’t mix at all because they have such vastly different properties. Supersymmetry allows us to interchange a fermion with a boson through the existence of hypothetical super partners, called sparticles. Each fermion is paired with a super-boson partner, and each boson has a super-fermion partner. Now they can be mixed via their super partners.

If it turns out to be true -- and thanks to the Large Hadron Collider, we could find evidence one way or the other in the coming years -- supersymmetry would provide physicists with a powerful calculating tool for understanding the most elusive mysteries of our universe, since everything would be connected to everything else through various kinds of dualities.

Okay, so dualities are powerful tools -- so what? How does understanding that help the average man on the street? That's what I love most about Amanda's choice: she finds a way to bring this high-falutin' concept down to earth:

In everyday language, duality means something very different. It is used to connote a stark dichotomy: male and female, east and west, light and darkness. Embracing the physicist's meaning of duality, however, can provide us with a powerful new metaphor, a one-stop shorthand for the idea that two very different things might be equally true. As our cultural discourse is becoming increasingly polarized, the notion of duality is both more foreign and more necessary than ever. If accessible in our daily cognitive toolkit, it could serve as a potent antidote to our typically Boolean, two-valued, zero-sum thinking — where statements are either true or false, answers are yes or no, and if I'm right, then you are wrong. With duality, there's a third option. Perhaps my argument is right and yours is wrong; perhaps your argument is right and mine is wrong; or, just maybe, our opposing arguments are dual to one another.

I like her analogy. I'm always looking for new ways to find some shreds of common ground with those whose views are different -- often radically so -- from my own... while still allowing both parties to be true to their principles. It's harder than it seems. For example, if you find yourself arguing with someone who sees the world starkly in black and white, they'll accuse you of seeing everything in tepid shades of gray, with the implicit assumption that this is an inferior worldview. Forget about trying to expand on their chosen metaphor and insisting your world is a kaleidoscope of color; they'll just sneer at you for being trite (as if a black and white worldview is, like, deep?).

Personally, I view the world as emergent and complex: it may start with a few very simple rules, but things become tangled and murky and impossibly complicated in very short order. That is the essence of the human condition. Embracing dualities is one way to deal with those inevitable complications and find some common ground. I'll let Amanda have the last word:

That's not to say that we ought to descend into some kind of relativism, or that there are no singular truths. It is to say, though, that truth is far more subtle than we once believed, and that it shows up in many guises. It is up to us to recognize it in all its varied forms.

Comments

Love this article and the point it makes (though it took me several readings to grasp the duality concepts). I agree that the answer to our current quagmire of red state / blue state thinking may lie in accepting the inevitable existence of both states (pardon the possible pun there). Meanwhile I need to get back to the fascinating book I am reading(The Calculus Diaries, no sucking up intended), thanks again for the great post.

Arguably one of the best posts on this blog. On another note, The Calculus Diaries is the best calculus self help book out there. I only wish I'd known of the book before I took my calculus class last semester. Great job on the blog too!

Physics Cocktails

Heavy G

The perfect pick-me-up when gravity gets you down.
2 oz Tequila
2 oz Triple sec
2 oz Rose's sweetened lime juice
7-Up or Sprite
Mix tequila, triple sec and lime juice in a shaker and pour into a margarita glass. (Salted rim and ice are optional.) Top off with 7-Up/Sprite and let the weight of the world lift off your shoulders.

Any mad scientist will tell you that flames make drinking more fun. What good is science if no one gets hurt?
1 oz Midori melon liqueur
1-1/2 oz sour mix
1 splash soda water
151 proof rum
Mix melon liqueur, sour mix and soda water with ice in shaker. Shake and strain into martini glass. Top with rum and ignite. Try to take over the world.